We introduce new partial orders on the set $Sn+$ of positive definite matrices of dimension $n$ derived from the affine-invariant geometry of $Sn+$. The orders are induced by affine-invariant cone fields, which arise naturally from a local analysis of the orders that are compatible with the homogeneous geometry of $Sn+$ defined by the natural transitive action of the general linear group $GL(n)$. We then take a geometric approach to the study of monotone functions on $Sn+$ and establish a number of relevant results, including an extension of the well-known L"owner-Heinz theorem derived using differential positivity with respect to affine-invariant cone fields.